It is not unusual in the world of extra dimensional physics to come across some closed-minded individual who is quite insistent that he lives in a universe with three (space) dimensions. Such people are generally unmoved by the standard physics response “But string theory is very pretty and anomaly cancellation requires it to live in a minimum of ten dimensions”, often retorting with something along the lines of “Look, count the buggers, you idiot: [gestures left and right] One, [gestures back and forth] Two, [and up and down] Three!” At this point it is standard practice to roll one’s eyes, smile patronisingly and point the ignorant fool in the direction of Edwin Abbott’s Flatland — the story of the inhabitants of a two dimensional universe suffering similar delusions of low dimensionality. Having sent my fair share of people on such a path, I thought it was time I actually read the book myself.

That Abbott thought to worry about such issues when he wrote the book in 1884 is quite amazing. This was exactly a century before the first string theory revolution made extra dimensions fashionable, forty years before Kaluza and Klein first suggested that considering extra spatial dimensions might lead to meaningful physics, and even a few decades before Einstein made his modest expansion of the dimensionality of our universe — from a three dimensional space to a four dimensional spacetime. Even pure mathematicians were only beginning to realise the value of considering such an abstract concept, and then only on the understanding that these were esoteric exercises unrelated to our real world. On these grounds alone Abbott’s book deserves to be remembered.

But what is all the more remarkable is that it is a hilarious read, providing a brilliantly biting satire of Victorian (and more recent) social structure and values and the bizarre logic on which it was based. Our narrator is A. Square (yep, he’s a square) — a rather pompous inhabitant of Flatland. In the first half of the story he introduces us to the laws, traditions, history and difficulties of life in this 2D universe while in the second half he tells of his travels to other worlds of both lower and higher dimensionality.

Flatland, we are told, is a place governed by the Laws of Nature: It is a law of nature that the son of an equilateral triangle will be a square, his son a pentagon and so on, with each generation rising a step up the social hierarchy until they eventually reach the pinnacle, being declared a circle. It is the natural law that regularity is sacrosanct and hence any irregular shaped being is destroyed at birth. An obvious corollary is that the lowest (well, second lowest, but more of that in a moment) rank in this caste system are the ‘Isosceles’ who have only two of their three sides equal and from the most acute of whom ‘a mere touch from the vertex … brings with it danger of death’. The Isosceles have no hope of climbing the polygonal ladder of opportunity from one generation to the next. The extent of their hopes is that their son might have an acute angle half a degree bigger and that their far distant descendents might one day be declared equilateral. It is a law of nature that Isosceles shall be soldiers and workmen, equilaterals shall be the middle class, squares and pentagons the professional class, polygons the nobility and circles the priests. It is the natural law that marriage shall only be between a man and a woman — oh no, my mistake, that’s Howard’s Australia. It is depressingly easy to confuse the two.

As you have no doubt guessed, the one class lower than the Isosceles are the straight lines or, to give them their other name, women. It is here that Abbott’s critique of Victorian society is most harsh, at least to a reader a century or so later, informed by the women’s suffrage and feminism. It is also when he is at his most hilarious, completely unconstrained by one of the unfortunate by products of such movements — political correctness. It is very hard to imagine such unswervingly sexist satire being written in our ‘enlightened’ society:

For as they have no pretensions to an angle, being inferior in this respect to the very lowest of the Isosceles, they are consequently wholly devoid of brain-power, and have neither reflection, judgment nor forethought, and hardly any memory.

We are also told of the difficulties of life in Flatland. Women, being lines, can make themselves invisible at will. Both they and acute Isosceles can inflict fatal wounds with the slightest touch. This situation is made all the more precarious since there is no colour in Flatland and its inhabitants, having only one eye, can’t perceive depth. As a result, everyone, be they Square, Circle or Dodecagon, looks like a line. For the lower classes, this means that recognition is achieved by (very careful so as to avoid impaling) touch. The higher classes make use of the fog which is universal in Flatland to recognise by the more refined art of sight, with beings with less sides disappearing more rapidly into the fog.

Things were not always thus in Flatland. Mr Square tells of Chromatistes, a legendary figure who discovered colour. It was not long before almost everyone was wearing it and using it to distinguish their front from their back, or to label themselves as triangle or pentagon. But with this new invention the higher polygons no long had the advantage that recognition by sight gave them, leading to a popular revolution and a decline of the arts. In this section, the book reminded me of that other great satire where the place of humans is taken by lower lifeforms — Animal Farm. Eventually, of course, the revolution is defeated and Flatland is returned to the dull, monochrome place it once was, although the natural laws once more held true so all was as it should be.

In the second half of the book we also get to see what life is like in other dimensionalities. First our narrator dreams of Lineland, a place where you remain next to the same two individuals for your entire life, for passing is impossible. He meets the King of this world (who, being a line, he initially mistakes for a woman) and attempts to explain the existence of Flatland but to no avail. On waking he is astounded to encounter a circle who is able to change his size, at first accusing him of using some sort of magic. When the ‘circle’ explains he is in fact not a circle but some generalisation of a circle, called a sphere, passing through the 2D world of Flatland, A. Square will not have it. It is only once the Sphere takes him to Spaceland and shows him such wonders as a cube that he can finally see it. However when he asks the Sphere what the generalisation of a cube to 4D looks like, the sphere reacts with incredulity at such an absurd suggestion, throwing the Square back down to Flatland. And it is there that he is destined to live out his days, the beauty that is a cube just a fading memory. In a final vision, the sphere appears to him and takes him to see Pointland — home to a single point that is the entirety of its universe, speaking of itself in the third person with self-important glee:

“It fills all Space,” continued the little soliloquizing Creature, “and what It fills, It is. What It thinks, that It utters; and what It utters, that It hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it is the One, and yet the All in All. Ah, the happiness ah, the happiness of Being!”

And this is the essential lesson of Flatland — we are all narrow-minded, it is just a question of the number of dimensions in which our minds are narrow.

(Since it is out of copyright, the book is available online here.)

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This entry was posted by the illustrious Andrew Coulthurst on Monday, February 13th, 2006. It concerns such pressing issues as books. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.